Exact Wave Spacetimes

Exact wave spacetimes separate invariant radiative geometry from coordinate appearance. In this chapter a parallel wave has a covariantly constant null direction, a pp-wave additionally has flat screen-space curvature, a plane wave has a quadratic transverse profile, and the Kundt family is defined by vanishing null optical scalars. Smooth sandwich profiles then provide the controlled bridge to impulsive curvature, while colliding-wave solutions expose genuinely nonlinear focusing.

Helpful background. Einstein’s Equation supplies vacuum and sourced field equations; The Riemann Curvature Tensor fixes tidal-curvature conventions; Frames, Coframes, and Anholonomy supplies null frames; Affine and Variational Geodesics supplies optical scalars, Jacobi fields, and focusing; Exact Solutions of Einstein’s Equations supplies the volume’s identification method; and The Wave Equation and Characteristics supplies characteristics and finite-profile language.

Enter this chapter

The recurring local metric is the Brinkmann form

$$ds^2=-2\,du\,dv+\delta_{ij}dx^idx^j +H(u,x)\,du^2.$$

Its curvature is governed by a transverse Hessian,

$$R_{uiuj}=-\frac12\partial_i\partial_jH.$$

This formula immediately distinguishes curvature from removable constant and affine pieces of $H$. It also fixes the geodesic-deviation sign and reduces the vacuum field equation to $\Delta_\perp H=0$. When $H=A_{ij}(u)x^ix^j$, the Hessian is independent of transverse position and the pp-wave is a plane wave.

The pp-wave, plane-wave, and colliding-wave families and their invariant interpretations are surveyed in Bičák 2000, §8, pp. 66–76, Open PDF. The distributional impulse pages use the regularization and coordinate comparisons in Podolský 2002, §§3.1–3.3, arXiv manuscript pp. 7–13, Open PDF.

No claim in the chapter equates an exact wave with an astrophysical waveform. Plane waves are transversely infinite idealizations, pp-waves can have poor global behavior, and radiative interpretation depends on a controlled frame and domain. The gain is different: every relation between profile, curvature, optics, and probe response can be checked exactly.

From local geometry to finite wave trains

pp-Wave Spacetimes begins with the invariant condition $\nabla_a k^b=0$, distinguishes the broader parallel-wave class from the additional flat-screen pp-wave condition, and then adopts the standard transversely flat local form. It derives the transverse Poisson equation, identifies the plane-wave subclass, and gives the nonflat VSI pp-wave as a decisive counterexample to scalar-invariant classification. A parallelly propagated tidal frame, not an oscillatory coordinate coefficient, detects the wave.

Plane Waves in Brinkmann and Rosen Coordinates constructs the transformation $x=E(u)X$ from the Jacobi matrix $E''=AE$. Rosen coordinates follow a geodesic congruence and fail when $\det E=0$. Smooth Brinkmann curvature at that event proves that the Rosen degeneracy is a coordinate caustic rather than automatically a spacetime singularity.

The Kundt Family broadens the geometry to null geodesic congruences with zero expansion, shear, and twist. pp-waves form one narrow subclass. Degenerate Kundt metrics explain why scalar polynomial invariants can be incomplete: aligned positive boost-weight components vanish through all curvature derivatives even though Cartan frame data distinguish the metrics.

Sandwich Waves and Geodesic Response keeps $A_{ij}(u)$ smooth and compactly supported. Its exact transfer matrix maps incoming separation and relative velocity to outgoing data, cleanly separating displacement at the exit from a lasting velocity change. The conserved matrix Wronskian provides an independent analytic and numerical check.

Impulsive Gravitational Waves then takes a controlled thin-profile limit. The distributional Brinkmann description, a continuous locally Lipschitz metric, and Penrose cut-and-paste gluing are compared rather than mixed. Transverse geodesic velocity jumps in the singular chart correspond to $C^1$ passage in continuous coordinates; quoted matching data are meaningless unless the chart is named.

Finally, Colliding Plane Waves sets characteristic data on two null wavefronts. Each incoming region can be regular while the interaction equation drives the transverse area to zero. Generic exact families form curvature singularities; exceptional data can produce a Cauchy horizon, so area focusing and curvature blow-up must be checked separately.

A verification sequence

For any proposed exact wave geometry:

1. identify the preferred null congruence and compute its acceleration, expansion, shear, and twist;
2. calculate $R_{uiuj}$ in a declared frame and trace it to test the vacuum equation;
3. distinguish pp-wave, plane-wave, Kundt, and degenerate-Kundt conditions instead of using the names interchangeably;
4. propagate a Jacobi matrix and check its Wronskian;
5. if a profile is impulsive, derive it as a smooth sandwich limit or through a precise gluing construction;
6. test any apparent singularity with a regular frame, not only coordinates or polynomial scalars; and
7. state the global domain before making completeness or causal claims.

The same sequence prevents two opposite errors. A Rosen caustic can look singular although the Brinkmann curvature is regular; a VSI pp-wave can look invariantly indistinguishable from flat space if only complete scalar contractions are inspected.

Check your understanding

Consider $H(u,x,y)=a(u)+b_i(u)x^i+A_{ij}(u)x^ix^j$, with symmetric trace-free $A$. Which terms carry local vacuum gravitational curvature?

Solution

Only the quadratic Hessian contributes: $R_{uiuj}=-A_{ij}$. The functions $a(u)$ and $b_i(u)$ have zero transverse Hessian and can be removed locally by a shift of $v$ and a $u$-dependent transverse translation. Trace-free $A$ gives $R_{uu}=0$, while its two independent components describe the vacuum tidal polarizations in a chosen parallel screen frame.

References

• Bičák, Jiří. “Selected Solutions of Einstein’s Field Equations: Their Role in General Relativity and Astrophysics.” In Einstein’s Field Equations and Their Physical Implications, edited by Bernd G. Schmidt, 1–126. Lecture Notes in Physics 540. Berlin: Springer, 2000. DOI; Open PDF.
• Coley, Alan, Sigbjørn Hervik, and Nicos Pelavas. “Spacetimes Characterized by Their Scalar Curvature Invariants.” Classical and Quantum Gravity 26 (2009): 025013. DOI; Open PDF.
• Podolský, Jiří. “Exact Impulsive Gravitational Waves in Spacetimes of Constant Curvature.” In Gravitation: Following the Prague Inspiration, edited by Oldřich Semerák, Jiří Podolský, and Martin Žofka, 205–246. Singapore: World Scientific, 2002. DOI; Open PDF (arXiv manuscript).